Sylvester's determinant theorem

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In matrix theory, Sylvester's determinant theorem is a theorem useful for evaluating certain types of determinants. It is named after James Joseph Sylvester.

The theorem states that if A, B are matrices of size p × n and n × p respectively, then

det(I_p + AB) = det(I_n + BA),

where I_a is the identity matrix of order a.

This theorem is useful in developing a Bayes estimator for multivariate Gaussian distributions.

Sylvester (1857) stated this theorem without proof.